Question: Let $x^2+y^2=25$. What is the value of $\dfrac{d^2y}{dx^2}$ at the point $(4,3)$ ? Give an exact number.
Answer: Notice that the equation defines $y$ implicitly—we don't have an explicit expression for $y$ in terms of $x$. So we will have to use implicit differentiation. If we differentiate the equation once, we will be able to get an expression for $\dfrac{dy}{dx}$. Then we can differentiate the equation again to get an expression for $\dfrac{d^2y}{dx^2}$. Let's start by finding $\dfrac{dy}{dx}$ : $\dfrac{dy}{dx}=-\dfrac{x}{y}$ Now we can differentiate $\dfrac{dy}{dx}$ to find $\dfrac{d^2y}{dx^2}$ : $\dfrac{d^2y}{dx^2}=-\dfrac{x^2+y^2}{y^3}$ Finally, let's plug ${x=4}$ and ${y=3}$ into the expression we got: $\begin{aligned} \left.-\dfrac{ x^2+ y^2}{ y^3}\right\rvert_{({4},{3})}&=-\dfrac{({4})^2+({3})^2}{({3})^3} \\\\ &=-\dfrac{25}{27} \end{aligned}$ In conclusion, the value of $\dfrac{d^2y}{dx^2}$ at the point $(4,3)$ is $-\dfrac{25}{27}$.